4.2 Quasar microlensing

Each of these stars (or other compact objects, like black holes, brown dwarfs, or planets) acts as a “compact lens” or “microlens” and produces at least one new image of the source. In fact, the “macro-image” consists of many “micro-images” (Figure 9). But because the image splitting is proportional to the lens mass (see Equation (4)), these microimages are only of order a microarcsecond apart and can not be resolved. Various aspects of microlensing have been addressed after the first double quasar had been discovered [37, 38, 66, 90, 135, 169, 194].

Figure 9: “Micro-Images”: The top left panel shows an assumed “unlensed” source profile of a quasar. The other three panels illustrate the micro-image configuration as it would be produced by stellar objects in the foreground. The surface mass density of the lenses is 20% (top right), 50% (bottom left) and 80% (bottom right) of the critical density (cf. Equation (16)).

The surface mass density in front of a multiply imaged quasar is of order the “critical surface mass density”, see Equation (16). Hence microlensing should be occuring basically all the time. This can be visualized in the following way. If one assigns each microlens a little disk with radius equal to the Einstein ring, then the fraction of sky which is covered by these disks corresponds to the surface mass density in units of the critical density; this fraction is sometimes also called the “optical depth”.

The microlenses produce a complicated two-dimensional magnification distribution in the source plane. It consists of many caustics, locations that correspond to formally infinitely high magnification.

An example for such a magnification pattern is shown in Figure 10. It is determined with the parameters of image A of the quadruple quasar Q2237+0305 (surface mass density  = 0.36; external shear  = 0.44). Color indicates the magnification: blue is relatively low magnification (slightly demagnified compared to mean), green is slightly magnified and red and yellow is highly magnified.

Due to the relative motion between observer, lens and source, the quasar changes its position relative to this arrangement of caustics, i.e. the apparent brightness of the quasar changes with time. A one-dimensional cut through such a magnification pattern, convolved with a source profile of the quasar, results in a microlensed lightcurve. Examples for microlensed lightcurves taken along the yellow tracks in Figure 10 can be seen in Figure 11 for two different quasar sizes.

In particular when the quasar track crosses a caustic (the sharp lines in Figure 10 for which the magnification formally is infinite, because the determinant of the Jacobian disappears, cf. Equation (31)), a pair of highly magnified microimages appears newly or merges and disappears (see [26]). Such a microlensing event can easily be detected as a strong peak in the lightcurve of the quasar image.

Figure 10: Magnification pattern in the source plane, produced by a dense field of stars in the lensing galaxy. The color reflects the magnification as a function of the quasar position: the sequence blue-green-red-yellow indicates increasing magnification. Lightcurves taken along the yellow tracks are shown in Figure 11. The microlensing parameters were chosen according to a model for image A of the quadruple quasar Q2237+0305:  = 0.36,  = 0.44.

Figure 11: Microlensing Lightcurve for the yellow tracks in Figure 10. The solid and dashed lines indicate relatively small and large quasar sizes. The time axis is in units of Einstein radii divided by unit velocity.

In most simulations it is assumed that the relative positions of the microlenses is fixed and the lightcurves are produced only by the bulk motion between quasar, galaxy and observer. A visualization of a situation with changing microlens positions can be found in Figure 13 for three different values of the surface mass density:

•  = 0.2 in 13a.
•  = 0.5 in 13b.
•  = 0.8 in 13c. Update

The change of caustics shapes due to the motion of individual stars which can be looked at when clicking on one of the three panels of Figure 13 produces additional fluctuations in the lightcurve [106, 197].

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Figure 13: mpeg-Movie (1464 KB) a) Microlensing caustics for surface mass density  = 0.2. See the caustics move due to the microlenses changing positions. The sequences are described and analysed quantitatively in [197].

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Figure 13: mpeg-Movie (2712 KB) b) Microlensing caustics for surface mass density  = 0.5. See the caustics move due to the microlenses changing positions. The sequences are described and analysed quantitatively in [197].

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Figure 13: mpeg-Movie (2622 KB) c) Microlensing caustics for surface mass density  = 0.8. See the caustics move due to the microlenses changing positions. The sequences are described and analysed quantitatively in [197].

This change of caustics shapes due to the motion of individual stars produces additional fluctuations in the lightcurve [106, 197].

Microlens-induced fluctuations in the observed brightness of quasars contain information both about the light-emitting source (size of continuum region or broad line region of the quasar, brightness profile of quasar) and about the lensing objects (masses, density, transverse velocity). Hence from a comparison between observed and simulated quasar microlensing (or lack of it) one can draw conclusions about the density and mass scale of the microlenses. It is not trivial, though, to extract this information quantitatively. The reason is that in this regime of optical depth of order one, the magnification is not due to a single isolated microlens, but it rather is a collective effect of many stars. This means individual mass determinations are not just impossible from the detection of a single caustic-crossing microlensing event, but it does not even make sense to try do so, since these events are not produced by individual lenses⁸. Mass determinations can only be done in a statistical sense, by comparing good observations (frequently sampled, high photometric accuracy) with simulations. Interpreting microlensed lightcurves of multiply-imaged quasars allows to determine the size of the continuum emitting region of the quasar and to learn even more about the central engine [68, 83, 145, 198].

So far the “best” example of a microlensed quasar is the quadruple quasar Q2237+0305 [76, 80, 110, 134, 198, 200, 207]. In Figure 14 two images of this system are shown which were taken in 1991 and 1994, respectively. Whereas on the earlier observation image B (top) is clearly the brightest, three years later image A (bottom) is at least comparable in brightness. Since the time delay in this system is only a day or shorter (because of the symmetric image arrangement), any brightness change on larger time scales must be due to microlensing. In Figure 15 lightcurves are shown for the four images of Q2237+0305 over a period of almost a decade (from [109]). The changes of the relative brightnesses of these images induced by microlensing are obvious.

Figure 14: Two images of the quadruple quasar Q2237+0305 separated by three years. It is obvious that the relative brightnesses of the images change. Image B is clearly the brightest one in the left panel, whereas images A and B are about equally bright in the right panel. (Credits: Geraint Lewis.)

Figure 15: Lightcurves of the four images of Q2237+0305 over a period of almost ten years. The changes in relative brightness are very obvious. (Credits: Geraint Lewis.)